Classical Statistical Mechanics
A macrostate has N particles arranged among m volumes,
with Ni(i = 1 . . . m) particles in the ith volume. The total
number of allowed microstates with distinguishable particles is
N!
;
W = Qm
N
!
i
i
ln W = ln N ! −
m
X
ln Ni ! .
i
For a large number of particles, use Stirling’s formula
ln N ! = N ln N − N .
m
X
ln W = N ln N − N −
(Ni ln Ni − Ni) .
i
The optimum state is the macrostate with the largest possible
number of microstates, which is found by maximizing W , subject to the constraint that the total number of particles N is
fixed (δN = 0). In addition, we require that the total energy
be conserved. If wi is the energy of the ith state, this is
!
m
m
X
X
δ
w i Ni =
wiδNi = 0 .
i
i
With these constraints, the minimization is
"
!#
m
m
X
X
δ ln W − α
Ni − β
w i Ni
= 0.
i
m
X
i
i
[ln Ni − α − βwi] δNi = 0 .
Ni = αeβwi = αe−wi/kT ,
which is the familiar Maxwell-Boltzmann, or classical, distribution function.
Quantum Statistical Mechanics
In the quantum mechanical view, only within a certain phase
space volume are particles indistinguishable. The minimum
phase space is of order h3. Now denote the number of microstates per cell of phase space of volume h3 as Wi . Then the
number of microstates per macrostate is
Y
W =
Wi .
i
Note we have to consider both the particles and the compartments into which they are placed. If the ith cell has n compartments, there are n sequences of Ni + n − 1 items to be
arranged. There are n(Ni + n − 1)! ways to arrange the particles and compartments, but we have overcounted because there
are n! permutations of compartments in a cell, and the order
in which particles are added to the cell is also irrelevant (the
factor Ni! we had in the classical case). Thus
Y n (Ni + n − 1)! Y (Ni + n − 1)!
=
.
W =
Ni !n!
Ni ! (n − 1)!
i
i
Optimizing this, we find
X
δ ln W =δ
[(n + Ni − 1) ln (n + Ni − 1) − Ni ln Ni
i
− (n − 1) ln (n − 1) − ln αNi − βwiNi ]
X n + Ni − 1
=
ln
− ln α − βwi δNi = 0 ,
Ni
i
or
Ni = (n − 1) αewi/kT − 1
−1
.
In fact, this is the relevant expression when there is no limit
to the number of particles that can be put into the compartment of size h3, i.e., for bosons. Further, in the case when
bosons are photons, the condition δN does not apply, and the
factor α ≡ 1.
For fermions, only 2 particles can be put into a compartment, where 2 is the spin degeneracy. Thus, phase space is
composed of 2n half-compartments, either full or empty. There
are no more than 2n things to be arranged and therefore no
more than 2n! microstates. But again, we overcounted. For Ni
filled compartments, the number of indistinguishable permutations is Ni!, and the number of indistinguishable permutations
of the 2n − Ni empty compartments is (2n − Ni)!. In this case,
we therefore have
Y
(2n)!
W =
.
Ni ! (2n − Ni)!
i
As before, we optimize:
X
δ ln W =δ
[2n ln (2n) − (2n − Ni) ln (2n − Ni )
i
or
− ln αNi − βwiNi ]
X 2n − Ni
=
ln
− ln α − βwi δNi = 0 ,
Ni
i
−1
w
/kT
Ni = 2n αe i
+1
.
The quantity ln α can be associated with the negative of the
degeneracy parameter µ/T , where µ is the chemical potential,
of the system. The classical case is the limit of the fermion
or boson case when α → ∞, since in this case the ±1 in the
denominator of the distribution function does not matter. In
the boson case, also, α ≥ 1 since wi > 0 and Ni > 0. Bosons
become degenerate when α → 1. For photons, α = 1. In
the fermion case, there is no restriction on the value of α, and
fermions become degenerate when α → −∞.
Thermodynamics
The internal energy U is
X
U = TS − PV +
µi N i
i
and the first law is
dU = T dS − P dV +
This implies
V dP − SdT −
X
X
µi dNi .
i
Ni dµi = 0 .
i
The Helmholtz F and Gibbs G free energies are
X
F = U − TS ;
G=
µi N i .
i
dF = −SdT −P dV +
X
i
µi dNi ;
dG = V dP −SdT +
The thermodynamic potential Ω = −P V obeys
X
dΩ = −SdT − P dV −
Nidµi .
X
dNi .
i
i
The following are useful thermodynamic
∂U ∂U =T
= −P
∂S V,Ni
∂V S,Ni
∂F ∂F = −S
= −P
∂T V,Ni
∂V T,Ni
∂Ω ∂Ω = −S
= −P
∂T V,µi
∂V T,µi
relations:
∂U = µi
∂Ni S,V,Nj6=i
∂F = µi
∂Ni T,V,Nj6=i
∂Ω = −Ni
∂µi T,V,µj6=i
Then ∂P/∂T |V,µi = S/V and ∂P/∂µi |T,V,µj6=i = Ni/V .
Statistical Physics of Perfect Gases—Fermions
The energy of a non-interacting particle is related to its rest
mass m and momentum p by the relativistic relation
E 2 = m 2 c4 + p 2 c2 .
(1)
The occupation index is the probability that a given momentum state will be occupied:
E − µ −1
f = 1 + exp
(2)
T
for fermions, where µ = ∂/∂n|s is the chemical potential and
is the energy density.
Figure 1: E, µ and T are scaled by mc2.
When the particles are interacting, E generally contains an
effective mass and a potential contribution. µ corresponds to
the energy change when 1 particle is added to or subtracted
from the system. The entropy per particle is s. We will use
units such that kB =1; thus T = 1 MeV corresponds to T =
1.16 × 1010 K. The number and internal energy densities are
given, respectively, by
Z
Z
g
g
= 3 Ef d3p
(3)
n = 3 f d3p;
h
h
where g is the spin degeneracy (g = 2j +1 for massive particles,
where j is the spin of the particle, i.e., g = 2 for electrons,
muons and nucleons, g = 1 for neutrinos). The entropy can be
expressed as
Z
g
ns = − 3 [f ln f + (1 − f ) ln (1 − f )] d3p
(4)
h
and the thermodynamic relations
∂ (/n)
| = T sn + µn − (5)
P = n2
∂n s
gives the pressure. Incidentally, the two expressions (Eqs. (4)
and (5)) are generally valid for interacting gases, also. We also
note, for future reference, that
Z
∂E 3
g
P = 3 p
f d p.
(6)
∂p
3h
Thermodynamics gives also that
∂P ∂P n=
ns =
(7)
;
.
∂µ T
∂T µ
Note that if we define degeneracy parameters φ = µ/T and
ψ = (µ − mc2)/T the following relations are valid:
∂P
∂P ∂P ∂ | ;
P = −+n +T
= ns+nφ;
= ns+nψ.
∂n T
∂T n
∂T φ
∂T ψ
(8)
In many cases, one or the other of the following limits may
be realized: extremely degenerate (φ → +∞), nondegenerate
(φ → −∞), extremely relativistic (p >> mc), non-relativistic
(p << mc).
Non-relativistic
In this case, one expands Eq. (3) in the limit p << mc.
Defining x = p2/(2mT ) and ψ = (µ − mc2)/T , one has
Z
g (2mT )3/2 ∞ x1/2dx
g (2mT )3/2
n=
F1/2 (ψ) (9)
≡
3
3
x−ψ
2
2
4π h̄
4π h̄
0 1+e
gT (2mT )3/2
2
= nmc +
F3/2 (ψ) .
3
2
4π h̄
(10)
Here, Fi is the usual Fermi integral which satisfies the recursion
dFi (ψ)
= iFi−1 (ψ) .
dψ
2
2
P =
− nmc ;
3
s=
5F3/2 (ψ)
3F1/2 (ψ)
(11)
− ψ.
Fermi integrals for zero argument satisfy
−i
Fi (0) = 1 − 2
Γ (i + 1) ζ (i + 1) ,
(12)
(13)
where ζ is the Riemann zeta function. Note that Fi (0) −−−→ i!.
Fi (ψ) may be expanded around ψ = 0 with
Fi (ψ) = Fi (0) + iFi−1 (0) ψ +
i→∞
i (i − 1)
Fi−2 (0) ψ 2 + · · · . (14)
2
Since F0(ψ) = ln(1 + eψ ), Fermi integrals with integer indices
less than 0 do not exist. The recursion Eq. (11) can be employed to define non-integer negative indices, however.
i
Fi (0)
i
Fi (0)
-7/2
-5/2
-3/2
-1/2
0
1/2
1
0.249109
0.2804865
-1.347436
1.07215
0.693147
.678094
0.822467
3/2
2
5/2
3
4
5
1.152804
1.803085
3.082586
5.682197
23.33087
118.2661
ln(2)
π 2/12
7π 4/120
31π 6/252
a. Non-degenerate and non-relativistic: In this limit,
using the expansion
∞
X
(−1)n+1 enψ
,
Fi (ψ) = Γ (i + 1)
i+1
n
ψ → −∞
(15)
s = 5/2 − ψ.
(16)
n=1
we find
n=g
mT 3/2 ψ
e ,
2πh̄2
P = nT,
b. Degenerate, non-relativistic: In this limit, we use
the Sommerfeld expansion
∞
ψ i+1 X (i + 1)!
Fi (ψ) =
i+1
(i + 1 − 2n)!
n=0
2n
π
Cn ,
ψ
ψ → ∞ (17)
Some values for the constants Cn are C0 = 1, C1 = 1/6, C2 =
7/360, and C3 = 31/15120. We find
"
#
2
3/2
1 π
g (2mψT )
1
+
n=
+··· ,
8 ψ
6π 2h̄3
"
#
2
5 π
2nψT
(18)
1+
+ ··· ,
P =
5
12 ψ
π2
s=
+ ···.
2ψ
Extremely relativistic
This case corresponds to setting the rest mass to zero. Eqs. (3)
and (5) become
3
g
T
n= 2
F2 (φ) ,
h̄c
2π
gT T 3
(19)
P = = 2
F3 (φ) ,
3 6π
h̄c
4F (φ)
− φ.
s= 3
3F2 (φ)
The above limiting expressions for the Fermi integrals may be
used in these expressions.
a. Extremely relativistic and non-degenerate: Use
of the expansion Eq. (15) results in
g T 3 φ
n= 2
e ,
h̄c
π
"
(20)
3 #
2
π n h̄c
= 4 − φ.
P =nT,
s = 4 − ln
g
T
b. Extremely relativistic and extremely degenerate: The expansion Eq. (17) gives
"
#
2
π
g
µ 3
1+
+ ··· ,
n= 2
φ
6π h̄c
"
#
2
nµ
π
(21)
P =
1+
+ ··· ,
4
φ
π2
s = +···
φ
Extremely degenerate
This case corresponds to φ >> 0. It is useful to define the
Fermi momentum pf for which the occupation index f = 1/2,
q
i.e., where µ = Ef = m2c4 + p2f c2. In terms of the parameter
x = pf /mc, we have
p
µ = mc 1 + x2.
2
(22)
In the case φ → ∞, Eq. (2) becomes a step function, with f = 1
for E ≤ µ; f = 0 for E > µ.
8A 3
x ,
mch2 p
i
2
−1
2
P = A x 2x − 3
1 + x + 3 sinh x ,
h p
i
2
2
3
−1
2
− nmc = A 3x 2x + 1
1 + x − 8x − 3 sinh x ,
n=
s = 0,
where A = (gmc2/48π 2)(mc/h̄)3.
(23)
Non-degenerate
This case corresonds to φ << 0. Because pair creation is
often important in this case, we delay detailed discussion of
limiting formulae for a later section. If pairs are neglected,
results may be expressed in terms of Bessel functions:
mc 3 T
2
φ
n=
e K2 mc /T ,
h̄
3mc2
P = nT,
mc 3 T
φ
2
2
e [−K1 mc /T +
− nmc =
(24)
h̄
3
3T /mc2 − 1 K2 mc2/T ],
2
2
mc K1 mc /T
− φ.
s=4−
T K2 mc2/T
General Comments About Fermions
Convenient scalings for electrons are achieved using
g m c 3
e
−9 fm−3
nc =
=
1.76
×
10
h̄
2π 2
2
mec =0.511 MeV
Fermions become relativistic under non-degenerate conditions when T > mc2 (T > 5 × 109 K for electrons) for any
density, and, under degenerate conditions, when pf c > mc2
(ρYe > 2 × 106 g cm−3 for electrons) for any temperature.
Here, ρ is the baryon density, and the number of electrons per
baryon is Ye. n(≡ ne) = ρNoYe . No is Avogadro’s number.
ψ ' 0 demarks the degenerate and non-degenerate regions
under all relativity conditions.
n=
ρYe
n=
ρYe
(2mT )3/2
g
F1/2 (0) ;
4π 2h̄3
3/2
T
−3
g
cm
' 2 × 106
5 × 109 K
3
T
g
F2 (0) ;
2π 2 h̄c
3
T
' 2 × 106
g cm−3
9
5 × 10 K
non − relativistic;
relativistic
separate the degenerate from the non-degenerate regions.
Interacting baryons are far more complicated. At subnuclear densities (ρ < ρo ≡ 2.7 × 1014 g cm−3) they cluster into
nuclei with internal densities near ρo. The nuclei themselves
are dilute, comprising a non-degenerate, non-relativistic gas,
but with a strong Coulombic (lattice) interaction. At very
high temperatures, the nuclei dissociate. Above ρo, nuclear interactions and degeneracy effects dominate. Baryons become
relativistic at a density (mbaryon /melectron )3 times higher than
the electrons, or about 1016 g cm−3. This is above the transition density to quark matter. At these densities, quarks can
be approximated as a perfect gas due to asymptotic freedom.
Fermion–Antifermion particle pairs
Under conditions found in the evolution of very massive
stars, the temperature may be high enough to produce electronpositron pairs, while the electrons are non-relativistic. During
gravitational collapse a degenerate neutrino-antineutrino gas
forms when densities large enough to trap neutrinos on dynamical time scales are reached (ρ > 1012 g/cm3). For particleantiparticle pairs in equilibrium, µ+ = −µ−. The net difference
of particles and anti-particles and the total pressure are
Z
1
1
4πg ∞ 2
p
−
n =n+ − n− = 3
dp,
(E−µ)/T
(E+µ)/T
h
1
+
e
1
+
e
Z0
4πg
1
∂E
1
P =P+ + P− = 3 p3
dp.
+
(E+µ)/T
∂p 1 + e(E−µ)/T
3h
1+e
(25)
Thus, when pairs are included, and n is positive, µ ≡ µ+
must be positive, i.e., there will not be cases involving extreme
non-degeneracy. However, pairs will never be important whenever µ/T >> 0, that is, under extremely degenerate conditions.
With the substitutions x = pc/T , z = mc2/T , we may write
Z ∞
T 3
x2
√
dx,
sinh φ
2
2
h̄c
0 coshφ + cosh z + x

√
3 Z ∞
2
2
− z +x
gT T
cosh
φ
+
e
x4

 dx.
√
√
P = 2
2
2
2
2
h̄c
6π
z + x cosh φ + cosh z + x
0
g
n= 2
2π
(26)
a. Extremely relativistic case: µ >> mc2 or T >> mc2.
This applies to neutrinos. With µ = µ+ = −µ−, i.e., z → 0,
3 h µ i
g
T
µ
− F2 −
n = n + − n− = 2
F2
h̄c
T
T
2π
"
#
g µ 3
πT 2
= 2
1+
;
µ
6π h̄c
µ i
gT T 3 h µ + F3 −
F3
/3 = P = P+ + P− = 2
h̄c
T
T
6π
"
4 #
2
µ 3
gµ
7 πT
πT
=
+
;
1
+
2
µ
15 µ
24π 2 h̄c
"
2 #
2
gT µ
7 πT
s=
.
1
+
3
15
µ
6n (h̄c)
(27)
These expressions are exact. The exponential terms ignored
in the Sommerfeld expansion of the +µ/T Fermi integral are
exactly canceled by those of the −µ/T Fermi integral. The pair
Fermi integral
Gi (η) ≡ Fi (η) + (−1)i+1 Fi (−η)
i≥0
obeys the same recursion formula as Fi(η) for i ≥ 1.
n(µ) is a cubic in µ, which can be inverted:
1/3
1/2
+t
,
µ = r − q/r,
r = q 3 + t2
(28)
where t = 3π 2(h̄c)3n/g and q = (πT )2/3. For T → ∞, one has
µ → 6n(h̄c)3/gT 2 → 0+. For all µ and T the adiabatic index
2 −1
d ln P T dP
d
d ln P = 4/3. (29)
Γ1 =
=
+
d ln n s
d ln n T P dT n dT n
One may include the lowest order corrections for finite rest
mass by expanding the integrands of Eq. (3) and using the
recursion relations for the Fermi integrals:
3
g µ 3
1 + µ−2 π 2T 2 − m2c4 ,
n= 2
2
6π h̄c
π2T 2 7
gµ µ 3
2 T 2 − m 2 c4
−2 2π 2 T 2 − 3m2c4 +
π
1
+
µ
P =
15
24π 2 h̄c
µ4
π2T 2 7
gµ µ 3
1
= 2
π 2 T 2 − m 2 c4
1 + µ−2 2π 2T 2 − m2c4 + 4
15
3
8π h̄c
µ
gT µ2
7 2 2 1 2 4
−2
s=
π T − m c
.
1
+
µ
3
15
2
6n (h̄c)
(30)
The relativistic relationship = 3P no longer holds. Interestingly, the cubic relationship between µ and n is preserved in
this approximation, and the solution is still given by Eq. (28) if
we simply redefine q = (πT )2/3 − m2c4/2. Including the finite
rest mass terms lowers Γ1 below 4/3:
Γ1 =
4
3
1−
5
11
2
mc2
πT
!
(31)
when photons (see below) are also included.
b. Non-relativistic case: µ << mc2 and T << mc2.
In the degenerate case, T → 0, µ → (mc2)+ and pairs are of
negligible importance. We can use the non-relativistic, degenerate formulas already obtained for particles alone. At higher
temperatures, µ reaches a maximum, and then decreases, eventually becoming less than mc2, so that µ0 < 0. The gas is thus
at most only partially degenerate when pairs are present and
n = n+ − n− > 0. The non-degenerate expansion yields
mT 3/2 [±µ−mc2]/T
n± ' g
.
e
2
2πh̄
Noting that n = n+ − n− and
3
mT
−2mc2/T ≡ n2 ,
e
n+ n− = g 2
1
2πh̄2
we can instead write
1/2
n
n 2
.
+ n21
n± = ∓ +
2
2
2 1/2
T,
n + 4n1
2
P = (n+ + n−) T =
3
= (n+ + n−) mc2 + T ,
2
5 mc2 (n+ + n−) µ
s =
+
− ,
2
T
n
T

!1/2 
n2
n

.
µ = T ln
+
+1
2
2n1
4n1
(32)
(33)
(34)
(35a)
(35b)
(35c)
(35d)
Pairs are important in the non-relativistic case when n ≤ n1.
Including photon pressure (see below), in the case when n <<
n1, one has
"
#
7/2
2
2
15 2mc
4
1−
e−mc /T .
(36)
Γ1 '
3
32
πT
Thus Γ1 reaches a minimum value (1.02) when T = 27 mc2, and
is always less than 43 . The creation of a pair costs an energy of
2mc2 which is non-negligible in the non-relativistic case.
c. Non-degenerate case: ψ < 0
Consider the region for which cosh φ − 1 << cosh z. Since
µ = T φ cannot be negative when pairs are included, the gas
is at most partially degenerate in this non-degenerate limit.
Expanding the cosh φ in the denominator terms to lowest order
in cosh φ − cosh z,
Z ∞
x2dx
−3
√
n = ncz sinh φ
,
2
2
0 1 + cosh z + x
Z ∞
x2dx
−3
√
P = ncT z [(cosh φ − 1)
0 1 + cosh z 2 + x2
Z
1
2 ∞ x4dx
√
√
],
+
(37)
3 0
x2 + z 2 1 + e x2+z 2
Z ∞
2
2
x x + z dx
√
= ncT z −3[(cosh φ − 1)
1
+
cosh
z 2 + x2
0
√
Z ∞ 2 2
x z + x2
√
dx],
+2
2 +x2
z
0 1+e
where nc = (g/2π 2)(mc/h̄)3 = 6×106 g cm−3 for g = 2. This is
an interesting approximation because, given n and T , one can
immediately evaluate µ or φ because they no longer appear
within the integrals. The integrals can be easily evaluated by
quadrature, with relatively few points, using Gauss-Laguerre
for z < 30 and Gauss-Hermite for z > 30.
When are pairs important?
In the relativistic case, n− = 0.1n+ is equivalent to F2 (φ) '
10F2(−φ)
√ or φ ' 0.9. In the non-relativistic case, we find
φ ' ln 10 or φ ' 1.15. φ ' 1 is the effective boundary.
The intrusion of this boundary into the NDNR region means
that there are actually five limiting cases when pairs are considered, as opposed to four when pairs are ignored. This is an
unfortunate complication.
Generalized Approximation
We explore here a technique invented by Eggleton, Faulkner
and Flannery (A& A 23, 325 [1973]) to bridge the limiting
regions for a fermion gas. It is essential to maintain thermodynamic consistency in this approximation. To include pairs we
simply apply the scheme separately to electrons and positrons.
The scheme establishes an analytic formula for the thermodynamic potential (or pressure) as an explicit function of chemical potential and temperature. Then n = T −1∂P/∂ψ; ns =
∂P/∂T − nψ; = T (∂P/∂T ) − P + nmc2. Density and temperature are inputs, so iteration is necessary to determine the
chemical potential. Johns, Ellis & Lattimer (ApJ 473, 1020
[1996]) improved the accuracy of the scheme and corrected the
behavior of the entropy in the degenerate limit.
The four limiting cases we have discussed are:
XX

4
(ψT
)
amn ψ −2m (ψT )−n



XX


5/2

(ψT
)
bmn ψ −2m (ψT )n
P
XX
=
4
ψ
nc mc2 
T e
cmn emψ T −n




XX

5/2 ψ
T e
dmn emψ T n
ER, ED : ψT >> mc2, ψ >> 1
NR, ED : ψT << mc2 , ψ >> 1
ER, ND : ψ << −1, T >> mc2
NR, ND : ψ << −1, T << mc2
(38)
where nc = (g/2π 2)(mc/h̄)3. The coefficients amn, bmn, cmn
and dmn (m, n ∈ 0 . . . ∞) can be determined from the limits.
The key is to find functions f (ψ), g(ψ, T ) such that Eq. (38)
can be rewritten as
 XX
4
0
−m −n

g
a
f
g
ER, ED : g >> 1, f >> 1

mn


XX


5/2

b0mnf −m g n NR, ED : g << 1, f >> 1
g
P
XX
=
4
ncmc2 
fg
c0mnf m g −n ER, ND : g >> 1, f << 1



XX


 f g 5/2
d0mnf m g n NR, ND : g << 1, f << 1.
(39)
This is possible provided that
X
(
2
ψ
rmψ −2m
ED :
ψ >> 1
X
f (ψ) =
(40)
ψ
mψ
e
sm e
ND :
ψ << −1
and
g (ψ, T ) =
(
tmψ −2m
X
T
umemψ .
ψT
X
ED :
ψ >> 1
ND :
ψ << −1
(41)
rmn, smn, tmn and umn are additional coefficients. Then
PM PN
Pmnf mg n
f
P
3/2
5/2
0
0
=
(42)
g (1 + g)
ncmc2 1 + f
(1 + f )M (1 + g)N
has the proper limits for any M, N ≥ 1 when f and g are either
large or small. Pmn are coefficients which are least squares fit
to a numerical evaluation of P . From Eq. (41), g ∝ T , and
p
2
g = T /mc
1+f
(43)
guarantees the right limiting behavior in Eq. (41).
√
On the other hand, it is also clear that ∂f /∂ψ is either f
(f → ∞) or f (f → 0). We choose
p
∂f /∂ψ = f / 1 + f /a,
(44)
where a is an adjustable parameter introduced by Johns, Ellis
& Lattimer (1996) which greatly improves the accuracy of the
scheme. Integrating this relation, we find the required relation
between ψ and f :
p
p
1 + f /a − 1
ψ = 2 1 + f /a + ln p
.
(45)
1 + f /a + 1
Explicitly evaluating the density, we find
1 ∂f ∂P ∂g ∂P 1 ∂P (46)
n=
=
+
;
T ∂ψ T
T ∂ψ ∂f g ∂f T ∂g f
PM P N
3/2
Pmnf m g n
fg
n
0
0
=p
×
M +1/2
N −3/2
nc
1 + f /a (1 + f )
(1 + g)
f
1 n
fg
3 N
1+m+
+ −M +
−
.
1+f 4 2
(1 + f ) (1 + g) 4
2
(47)
From P + U = T (∂P/∂T )ψ , where U = − nmc2 is the internal
energy density,
P M PN
Pmnf m g n
U
3/2
5/2
0
0
= f g (1 + g)
×
M +1
N
ncmc2
(1 + f )
(1 + g)
g
3
3
+n+
−N
.
2
1+g 2
(48)
Given n and T , we invert Eq. (47) to determine f ; g is trivially
found. The pressure is given by Eq. (42), the chemical potential
by Eq. (45), and the energy density by Eq. (48).
The entropy is found from s = n−1(∂P/∂T )ψ − ψ. A drawback of the Eggleton et al. scheme was that in the degenerate
limit, although the entropy
per particle has the correct asymp√
totic dependence 1/ f , the coefficient is not exact:
r r
π2 a
a
1 M PM −1,N
−−−−−−→
2+ −
+
ED, ER
f
a
a
aPM,N
f
M,N →∞ 2
s=
r 2ra

PM −1,0
8
a
5
1
M
π


+
−
+
. ED, NR
−−−−−−→

5 f 4 4a
a
aPM,0
f
M,N →∞ 4
(49)
For M, N = 2(3), Eqs. (49) have errors of 1.35 (0.0165)% and
0.254 (0.0637)%, respectively, for the ER and NR cases. The
original Eggleton et al. errors for these cases (they assumed
a = 1) are (2.4 (1.40)%, 1.0 (0.563)%), respectively.





From Eqs. (49), the corner values of Pmn should be:
Pmn
m=0
m=M −1
m=M
n=0
p
e2 π/32/a
n=N
e2/(2a)
5π 2 −40+(32M −8)/a
15a1/4
32/(15a5/4)
2π 2 −8+4(M −1)/a
3a
4/(3a2)
Johns et al. constrained the fit so that these corner values
and the ED entropies in Eq. (49) are exactly fulfilled. For
M = N = 3 we find an optimum fit for a = 0.433 with a
root-mean-square error of 8.1 · 10−5 and a maximum error of
3.0 · 10−4 at the fitting points. The coefficients Pmn are:
Pmn
n=0
n=1
n=2
n=3
m=0
m=1
m=2
m=3
5.34689
16.8441
17.4708
6.07364
18.0517
55.7051
56.3902
18.9992
21.3422
63.6901
62.1319
20.0285
8.53240
24.6213
23.2602
7.11153
To extend this scheme to include pairs, let the subscript +
refer to particles and − refer to antiparticles. Then
n = n+ (f+, g+) − n− (f− , g−)
p
2
where g± = (T /mc ) 1 + f± and
p
p
1 + f± /a − 1
ψ± = 2 1 + f±/a + ln p
.
1 + f± /a + 1
(50)
(51)
One solves the simultaneous equations
A = n − n+ (f+ , T ) + n− (f− , T ) = 0;
B = ψ+ (f+) + ψ− (f−) + 2mc2/T = 0,
(52)
where the second follows from µ− = −µ+. This is readily
handled, since the derivatives are analytic:
p
∂A/∂f± = ∓∂n±/∂f±;
∂B/∂f± = 1 + f±/a/f±. (53)
Boson Gas
The boson pressure and energy density are obtained by employing the same equations as for fermions, but using the Bose
distribution function −1
E−µ
fB = exp
,
(54)
−1
T
and a slightly different
entropy formula
Z
g
ns = − 3 [fB ln fB − (1 + fB ) ln (1 + fB )] d3p.
(55)
h
Since the occupation index cannot be negative, a free (noninteracting) Bose gas µ ≤ mc2. If µ → mc2, a “Bose condensate” appears and there will be a finite number of particles in
a zero-momentum state. Some limiting cases:
a. Extremely Non-degenerate:
In the non-degenerate limit, µ/T → −∞, the Bose and
Fermion distributions become indistinguishable, so the limits for thermodynamic quantities evaluated previously for the
Fermi gas apply.
b. Extremely Degenerate
For bosons, the “degenerate” limit is µ = mc2 or ψ = 0; the
number density is
Z ∞
p2
g
dp.
(56)
n=
2π 2h̄3 0 e(E−mc2)/T − 1
The integrals in this case can be written simply in terms of
zero argument Fermi integrals:
Z ∞
−1
xi
−i
Fi (0) = Γ (i + 1) ζ (i + 1) .
x − 1 dx = 1 − 2
e
0
(57)
Therefore we have the following additional limits:
i. Relativistic (T >> mc2, ψ = 0)
4g T 3
n=
F2 (0) ,
= 3P ;
6π 2 h̄c
gπ 2 T T 3
π4
4gT T 3
F3 (0) =
,
s=
' 3.601571.
P =
90
h̄c
15F2 (0)
21π 2 h̄c
ii. Non-relativistic (T << mc2, ψ = 0)
g (mT )3/2 F1/2 (0)
√
n= 2
,
π
h̄3
2−1
4gT (mT )3/2 F3/2 (0)
P =
,
2
3
3/2
3π
h̄
2 −1
3
= nmc2 + P ;
2
10 F3/2 (0) 21/2 − 1
s=
' 1.283781.
3 F1/2 (0) 23/2 − 1
Note that in these limits the entropy per boson is constant. The
location of the ψ = 0 trajectory in a boson density-temperature
plot is not far from the same curve for fermions.
c. Extremely Relativistic
In this case, we take m → 0, and we arrive at the simplest
bose gas, the photon gas, for which µγ = 0. With gγ = 2, one
obtains
3
π 2T T 3
γ = 3Pγ = T Sγ =
,
(58)
4
15 h̄c
which is (8/7g) times the value for a relativistic fermion gas.
Here S is the entropy density. In any regime where electronpositron pairs are important, the photon pressure is also important. In the non-degenerate, relativistic domain, the total
pressure from photons and electron-positron pairs is therefore
11Pγ /4, Under situations when neutrino pairs of all three flavors are trapped in the matter, the total pressure increases to
43Pγ /8. In the regime where electrons are degenerate, however,
photon pressure is negligible.
In the regime where the electrons are non-degenerate and
pairs are not important, the non-degenerate gas pressure of
nuclei must be included, and photon pressure may or not be
important. The pressure due to photons is important at lower
temperatures than pair pressure, owing to the expense of creating electron-positron pairs. Since the photon pressure is (8/7g)
times the relativistic non-degenerate pair pressure, the boundary to the region in which photon pressure is significant is simply obtained by a continuation of the straight line relativistic
boundary ρ ∝ T 3 to low densities and is akin to the line φ = 1
in the fermion-antifermion pair case.
Even excluding the contribution from electron-positron pairs,
the adiabatic index of a non-relativistic gas changes from 5/3
to 4/3 as the temperature is increased and the contribution of
radiation pressure increases. Denoting the fraction of the total
pressure due to gas pressure (assuming complete ionization) by
β,
32 − 24β − 3β 2
Γ1 =
.
(59)
24 − 21β
This ultimately sets an upper limit to the mass of main sequence stars.
Johns, Ellis & Lattimer (1996) extended the Eggleton et al.
scheme to handle a boson gas.